There is a physically relevant notion of non-complete AdS spacetimes, namely, those which are globally hyperbolic: they contain a closed Cauchy surface, which we always assume to be of genus at least 2. Mess [1]}, [2]} discovered striking analogies between the geometric properties of 3-dimensional Globally Hyperbolic Maximal Compact (GHMC) anti-de Sitter spacetimes, and those of quasifuchsian hyperbolic manifolds. For this reason, GHMC AdS spacetimes are now often called “quasifuchsian AdS spacetimes”, and we will follow this convention here.
